When it is about the Valois-Angoulême, I can go very far. 😁
It was the time of the Eighty Years' War. Dutch Protestant rebels were fighting the Habsburgs' authority. In OTL, this war ended up with the formation of the Dutch Republic.
Dealing with that would be very difficult for ATL Philip III's younger half-brother.
I don't know. Sweden is not England and Sigismund is not Mary.
I do not know Swedish History enough to know how plausible a come back to Catholicism was in this time.
I am not sure.
I thought Elisabeth-Magdalena was the most strategic choice because she was Sigismund I's most senior descendant, being his eldest daughter's eldest daughter.
However,
@Zygmunt Stary answered the Vasas had more significant position because of being descendants of Sigismund II's full-sister instead of his half-sister.
In any case, a marriage between Henry and Anna Vasa would have some problems.
When he came to Poland in 1574, she was too young to marry. He has to wait at least until 1580. It is not that long but, meanwhile, it is pretty sure Polish nobility would pressure him to marry Anna Jagiellon. After all, they elected him because Jean de Montluc had promised he would marry her. But more important: I am pretty sure many Polish noblemen wanted to prevent Henry to marry a younger Jagiellon descendant who may give him a son. Why? Simply to be sure Polish royalty would stay elective.
Sure, Henry can marry Anna Jagiellon and wait her death to marry Anna Vasa. However, he doesn't know how long Anna Jagiellon will live and he has no reason to assume Anna Vasa would not marry someone else before her aunt's death.
And, last but not least, it is pretty likely that Sigismund Vasa (or a son of him if Henry outlives him) will be candidate to Henry's succession. Against Henry's hypothetical son.
Would John III of Sweden approve a marriage that would weaken Sigismund's candidacy? I am not sure at all, especially considering John III had himself been candidate against Henry.
Elisabeth-Magdalena of Brandenbourg, in addition of being Sigismund I's most senior descendant, was in the right age and available to marry in 1574. Moreover, her brothers were already dead without issue, meaning there is no potential competitor on this side.
After the St Bartholomew's Day Massacre, an anonymous pamphlet claimed the Guises were male line descendants of Charlemagne and should replace the Valois on the throne of France. The Guises said they had no knowledge about that and did not want to take the throne. But, of course, they had to say that. Saying otherwise would've been open rebellion.
If Elisabeth wants to protect her son and preserve the throne for him, giving some power to the Guises is the last thing to do.
As for the Cardinal of Bourbon... Yeah... Maybe this one would be a suitable ally to Elisabeth.